contours will grow exponentially over time. The figure above shows the frame-by-frame evolution of a contour advected over several days. The figure to the right shows the length of this contour as a function of time. The link between exponential contour growth and positive Lyapunov exponents is easy to see. The rate of contour growth is given as: where is the path and the integral is performed over the length of the contour. Contour growth rates will approximate the average of the large Lyapunov exponents: Poincaré sections In chaotic advection, a fluid particle travels within a large region, and encounters other particles that were initially far from it. One can then consider that a particle is mixed with particles that travel within the same region. However, the region covered by a trajectory does not always span the whole fluid domain. Poincaré sections are used to distinguish regions of good and bad mixing. The Poincaré map is defined as the transformation transforms a point-like particle into the position of the particle after a time-interval T. Especially, for a time-periodic flow with period T, applying the map several times to a particle gives the successive positions of the particle period after period. A Poincaré section is built by starting from a few different initial conditions and plotting the corresponding iterates. This comes down to plotting the trajectories stroboscoped every T. As an example, the figure presented here (left part) depicts the Poincaré section obtained when one applies periodically a figure-eight-like movement to a circular mixing rod. Some trajectories span a large region: this is the chaotic or mixing region, where good mixing occurs. However, there are also two "holes": in these regions, the trajectories are closed. These are called elliptic islands, as the trajectories inside are elliptic-like curves. These regions are not mixed with the remainder of the fluid. For mixing applications, elliptic islands have to be avoided for two reasons : Fluid particles are unable to cross the boundaries of the islands (except by slow diffusion), resulting in segregation. Mixing inside these regions is not efficient because trajectories are closed and therefore not chaotic. Avoiding non-chaotic islands requires understanding the physical origin of these regions. Generally speaking, changing the geometry of the flow can modify the presence or absence of islands. In the figure-eight flow for instance, for a very thin rod, the influence of the rod is not felt far